6.1 - Introduction to Cube and Cube Roots

In this section, we will explore the concepts of cubes and cube roots, which are fundamental to understanding higher-level mathematics. These concepts are widely used in algebra, geometry, and real-world problem solving.

What is a Cube?

A cube of a number is the result of multiplying that number by itself three times. Mathematically, for any number $n$, the cube is denoted as:

Cube of $n$: $n^3$

This means that:

$n^3 = n \times n \times n$

For example:

• The cube of 2 is $2^3 = 2 \times 2 \times 2 = 8$.
• The cube of 3 is $3^3 = 3 \times 3 \times 3 = 27$.
• The cube of 4 is $4^3 = 4 \times 4 \times 4 = 64$.

What is a Cube Root?

A cube root is the reverse operation of cubing a number. The cube root of a number is the value that, when multiplied by itself three times, gives the original number. The cube root of a number $x$ is denoted as:

Cube root of $x$: $\sqrt[3]{x}$

Mathematically, the cube root of $x$ is the number $n$ such that:

$n^3 = x$

For example:

• The cube root of 8 is $\sqrt[3]{8} = 2$, because $2^3 = 8$.
• The cube root of 27 is $\sqrt[3]{27} = 3$, because $3^3 = 27$.
• The cube root of 64 is $\sqrt[3]{64} = 4$, because $4^3 = 64$.

Properties of Cube and Cube Roots

1. The cube of any positive number is always positive, and the cube of any negative number is always negative. For example:

• The cube of 5 is $5^3 = 125$ (positive).
• The cube of -5 is $(-5)^3 = -125$ (negative).

2. The cube root of a positive number is positive, and the cube root of a negative number is negative. For example:

• The cube root of 64 is $\sqrt[3]{64} = 4$ (positive).
• The cube root of -64 is $\sqrt[3]{-64} = -4$ (negative).

3. The cube of zero is $0^3 = 0$, and the cube root of zero is $\sqrt[3]{0} = 0$.

4. Cube and cube roots are inverse operations. That is, if $x = n^3$, then $\sqrt[3]{x} = n$, and if $x = \sqrt[3]{n}$, then $x^3 = n$.

Cube and Cube Roots in Algebra

Cubes and cube roots are often encountered in algebraic expressions. For example, if we are given the equation $x^3 = 27$, we can find the value of $x$ by taking the cube root of both sides:

By applying the cube root to both sides, we get:

$\sqrt[3]{x^3} = \sqrt[3]{27}$

This simplifies to:

$x = 3$

This method of finding the cube root is an essential part of solving cubic equations.